Чезаровское среднее

Чезаровское среднее (среднее по Чезаро) — среднее арифметическое частичных сумм первых членов заданной последовательности :

где  — частичные суммы ряда:

Названо в честь итальянского математика Эрнесто Чезаро.

Основной результат теории чезаровских средних — теорема Штольца — утверждает, что если существует предел последовательности частичных сумм , то также существует предел последовательности , и они равны:

.

Тем самым, операция взятия чезаровского среднего обладает свойством регулярности — сохраняет свойство сходимости последовательности и её предел. В то же время, существует множество примеров, когда исходная последовательность не имеет предела, а её чезаровские средние сходятся. (Например, последовательность .) Это позволяет использовать чезаровские средние как один из методов суммирования расходящихся рядов.

Ссылки

  • Кириллов А. А., Гвишиани А. Д. Теоремы и задачи функционального анализа.

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